Partition properties on \mathscr{P}_κλ

Abstract

Menas [{13}] showed there exist 2^{2^{λ^<κ}} normal ultrafilters on \mathscr{P}_κλ with the partition property if κ is 2^{λ^<κ}-supercompact. We first show that λ-supercompactness of κ implies the existence of a normal ultrafilter on \mathscr{P}_κλ with the partition property. We also prove by a similar technic that part^*(κ, λ) holds if κ is λ-ineffable with cf(λ)≥qκ. Note that Magidor [{11}] showed κ is λ-ineffable if part^*(κ, λ) holds, and we proved the converse under some additional assumption in [7].